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arXiv:math/0601636v1[math.AP]26Jan2006ERRORBOUNDSFORMONOTONEAPPROXIMATIONSCHEMESFORPARABOLICHAMILTON-JACOBI-BELLMANEQUATIONSGUYBARLESANDESPENR.JAKOBSENAbstract.Weobtainnon-symmetricupperandlowerboundsontherateofconvergenceofgeneralmonotoneapproximation/numericalschemesforpar-abolicHamiltonJacobiBellmanEquationsbyintroducinganewnotionofconsistency.Weapplyourgeneralresultstovariousschemesincludingfi-nitedifferenceschemes,splittingmethodsandtheclassicalapproximationbypiecewiseconstantcontrols.1.IntroductionInthisarticle,weareinterestedintherateofconvergenceofgeneralmonotoneapproximation/numericalschemesfortime-dependentHamiltonJacobiBellman(HJB)Equations.Inordertobemorespecific,theHJBEquationsweconsiderarewritteninthefollowingformut+F(t,x,u,Du,D2u)=0inQT:=(0,T]×RN,(1.1)u(0,x)=u0(x)inRN,(1.2)whereF(t,x,r,p,X)=supα∈A{Lα(t,x,r,p,X)},withLα(t,x,r,p,X):=−tr[aα(t,x)X]−bα(t,x)p−cα(t,x)r−fα(t,x).Thecoefficientsaα,bα,cα,fαandtheinitialdatau0takevaluesrespectivelyinSN,thespaceofN×Nsymmetricmatrices,RN,R,R,andR.Undersuitableassumptions(see(A1)inSection2),theinitialvalueproblem(1.1)-(1.2)hasaunique,bounded,H¨oldercontinuous,viscositysolutionuwhichisthevaluefunctionofafinitehorizon,optimalstochasticcontrolproblem.Weconsiderapproximation/numericalschemesfor(1.1)-(1.2)writteninthefol-lowingabstractwayS(h,t,x,uh(t,x),[uh]t,x)=0inG+h:=Gh\{t=0},(1.3)uh(0,x)=uh,0(x)inG0h:=Gh∩{t=0},Date:February8,2008.Keywordsandphrases.Hamilton-Jacobi-BellmanEquations,switchingsystem,viscositysolu-tion,approximationschemes,finitedifferencemethods,splittingmethods,convergencerate,errorbound.JakobsenwassupportedbytheResearchCouncilofNorway,grantno.151608/432.12BARLESANDJAKOBSENwhereSis,looselyspeaking,aconsistent,monotoneanduniformlycontinuousapproximationoftheequation(1.1)definedonagrid/meshGh⊂QT.Theap-proximationparameterhcanbemulti-dimensional,e.g.hcouldbe(Δt,Δx),Δt,Δxdenotingtimeandspacediscretizationparameters,Δxcanbeitselfmulti-dimensional.Theapproximatesolutionisuh:Gh→R,[uh]t,xisafunctiondefinedfromuhrepresenting,typically,thevalueofuhatotherpointsthan(t,x).Weassumethatthetotalschemeincludingtheinitialvalueiswell-definedonsomeappropriatesubsetofthespaceofboundedcontinuousfunctionsonGh.TheabstractnotationwasintroducedbyBarlesandSouganidis[3]todisplayclearlythemonotonicityofthescheme.OneofthemainassumptionsisthatSisnon-decreasinginuhandnon-increasingin[uh]t,xwiththeclassicalorderingoffunctions.Thetypicalapproximationschemeswehaveinmindarevariousfinitedifferencesnumericalscheme(seee.g.KushnerandDupuis[13]andBonnansandZidani[5])andcontrolschemesbasedonthedynamicprogrammingprinciple(seee.g.CamilliandFalcone[6]).However,forreasonsexplainedbelow,wewillnotdiscusscontrolschemesinthispaper.Theaimofthispaperistoobtainestimatesontherateoftheconvergenceofuhtou.Toobtainsuchresults,onefacesthedoubledifficultyofhavingtodealwithbothfullynonlinearequationsandnon-smoothsolutions.Sincetheseequationsmaybealsodegenerate,the(viscosity)solutionsareexpectedtobenomorethanH¨oldercontinuousingeneral.Despiteofthesedifficulties,inthe80’s,Crandall&Lions[10]providedthefirstoptimalratesofconvergenceforfirst-orderequations.WerefertoSouganidis[27]formoregeneralresultsinthisdirection.Fortechnicalreasons,theproblemturnsouttobemoredifficultforsecond-orderequations,andthequestionremainedopenforalongtime.Thebreakthroughcamein1997and2000withKrylov’spapers[20,21],andbynowthereexistsseveralpapersbasedonandextendinghisideas,e.g.[1,2,11,18,22,23].ThemainideaofKrylovisamethodnamedbyhimself“shakingthecoefficients”.Combinedwithastandardmollificationargument,itallowsonetogetsmoothsubsolutionsoftheequationwhichapproximatethesolution.Thenclassicalargumentsinvolvingconsistencyandmonotonicityoftheschemeyieldaone-sidedboundontheerror.Thismethodusesinacrucialwaytheconvexityoftheequationinu,Du,andD2u.Itismuchmoredifficulttoobtaintheotherboundandessentiallytherearetwomainapproaches.Thefirstoneconsistsofinterchangingtheroleoftheschemeandtheequation.Byapplyingtheaboveexplainedideas,onegetsasequenceofappropriatesmoothsubsolutionsoftheschemeandconcludesbyconsistencyandthecomparisonprinciplefortheequation.Thisideawasusedindifferentarticles,see[1,11,18,20,23].Here,thekeydifficultyistoobtaina“continuousdependence”resultforthescheme.EventhoughitisnowstandardtoprovethatthesolutionsoftheHJBEquationwith“shakencoefficients”remainclosetothesolutionoftheoriginalequation,suchtypeofresultsarenotknownfornumericalschemesingeneral.WementionherethenicepaperofKrylov[23]wheresuchkindofresultsareobtainedbyatrickyBernsteintypeofargument.However,theseresultsalongwiththecorrespondingerrorbounds,onlyholdforequationsandschemeswithspecialstructures.ERRORBOUNDS3Thesecondapproachconsistsofconsideringsomeapproximationoftheequationortheassociatedcontrolproblemandtoobtaintheotherboundeitherbyproba-bilisticarguments(asKrylovfirstdidusingpiecewiseconstantcontrols,[22,21])orbybuildingasequenceofappropriate“smoothsupersolution”oftheequation(see[2]where,asinthepresentpaper,approximationsbyswitchingareconsidered).Thefirstapproachleadstobettererrorboundsthanthesecondone